I don't know a lot about coyote hunting, just what I see on hunting shows.
But out of the 2 calibers you said, 223, and 243.
I would get the 243, because you can hunt deer with it also.
From what I have seen, get an inexspnive call, and hang a big feather from a tree limb, or from a stick, in the ground.
The feather will spin, and flip in the wind, and get the coyotes attention.
Or one of the electronic predator calls, with a spinning decoy on top.
On the coyote hunting shows, I have seen they call sporatically.
And get quiet when they see the coyote/coyotes.

What ever you do Good Luck, and post pictures.